Tables for Group Theory - UJI

1 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n Oxford University Press, 2006. All rights reserved. 1 Tables for Group Theory By P. W. ATKINS, M. S. CHILD, and C. S. G. PHILLIPS This provides the essential Tables (character Tables , direct products, descent in symmetry and subgroups) required for those using Group Theory , together with general formulae, examples, and other relevant information. Character Tables : 1 The Groups C1, Cs, Ci 3 2 The Groups Cn (n = 2, 3, .., 8) 4 3 The Groups Dn (n = 2, 3, 4, 5, 6) 6 4 The Groups Cnv (n = 2, 3, 4, 5, 6) 7 5 The Groups Cnh (n = 2, 3, 4, 5, 6) 8 6 The Groups Dnh (n = 2, 3, 4, 5, 6) 10 7 The Groups Dnd (n = 2, 3, 4, 5, 6) 12 8 The Groups Sn (n = 4, 6, 8) 14 9 The Cubic Groups: 15 T, Td, Th O, Oh 10 The Groups I, Ih 17 11 The Groups C v and D h 18 12 The Full Rotation Group (SU2 and R3) 19 Direct Products.

2 1 General Rules 20 2 C2, C3, C6, D3, D6, C2v, C3v, C6v, C2h, C3h, C6h, D3h, D6h, D3d, S6 20 3 D2, D2h 20 4 C4, D4, C4v, C4h, D4h, D2d, S4 20 5 C5, D5, C5v, C5h, D5h, D5d 21 6 D4d, S8 21 7 T, O, Th, Oh, Td 21 8 D6d 22 9 I, Ih 22 10 C v, D h 22 11 The Full Rotation Group (SU2 and R3) 23 The extended rotation groups (double groups): character Tables and direct product table 24 Descent in symmetry and subgroups 26 Notes and Illustrations: General formulae 29 Worked examples 31 Examples of bases for some representations 35 Illustrative examples of point groups: I Shapes 37 II Molecules 39 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n Character Tables Notes: (1) Sch nflies symbols are given for all point groups.

3 Hermann Maugin symbols are given for the 32 crystaliographic point groups. (2) In the groups containing the operation C5 the following relations are useful: 12121212(1 5 ) 1 618032 cos144(1 5 )0 618032 cos 721112 cos 722 cos1441 +=+== = = = +++ + =+=+=+ +=+= oLoLoo 1 Oxford University Press, 2006. All rights reserved. 2 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n 1. The Groups C1, Cs, Ci C1 (1) E A 1 Cs=Ch (m) E hA 1 1 x, y, Rzx2, y2, z2, xy A 1 1 z, Rx, Ryyz, xz Ci = S2 (1) E i Ag 1 1 Rx, Ry, Rzx2, y2, z2, xy, xz, yz Au1 1 x, y, z Oxford University Press, 2006. All rights reserved. 3 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n 2.

4 The Groups Cn (n = 2, 3,..,8) C2 (2) E C2 A 1 1 z, Rzx2, y2, z2, xy B 1 1 x, y, Rx, Ry yz, xz C3 (3) E C323C = exp (2 i/3) A 1 1 1 z, Rzx2 + y2, z2E **11 (x, y)(Rx, Ry) (x2 y2, 2xy)(yz, xz) C4 (4) E C4C234C A 1 1 1 1 z, Rzx2 + y2, z2B 1 1 1 1 x2 y2, 2xy E 1 i 1 i1i 1 i (x, y)(Rx, Ry) (yz, xz) C5E C525C 35C 45C = exp(2 i/5) A 1 1 1 1 1 z, Rzx2 + y2, z2E12*2**2211 (x,y)(Rx, Ry) (yz, xz) E22**2**2211 (x2 y2, 2xy) C6 (6)

5 E C6C3C223C 56C = exp(2 i/6) A 1 1 1 1 1 1 z, Rzx2 + y2, z2B 1 1 1 1 1 1 E1**1111 (x, y) (Rz, Ry) (xy, yz) E2**1111* (x2 y2, 2xy) Oxford University Press, 2006. All rights reserved. 4 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n 2. The Groups Cn (n = 2, 3,..,8) ( ) C7E C727C 37C 47C 57C 67C = exp (2 i/7) A 1 1 1 1 1 1 1 z, Rzx2 + y2, z2E123*3*2**2*33211 (x, y) (Rx, Ry) (xz, yz) E22*3*3*2*23**3211 (x2 y2, 2xy) E33*2*2*3**3*22311 C8E C8C4C234C 38C 58C 78C = exp (2 i/8) A 1 1 1 1 1 1 1 1 z, Rzx2 + y2, z2B 1 1 1 1 1 1 1 1 E1**1 i1i1i1 i* (x, y) (Rx, Ry) (xz, yz) E21 i111i ii1i 111 i i i (x2 y2, 2xy) E3**1 i1i1i1 i Oxford University Press, 2006.

6 All rights reserved. 5 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n 3. The Groups Dn (n = 2, 3, 4, 5, 6) D2 (222) E C2(z) C2(y) C2(x) A 1 1 1 1 x2, y2, z2BB11 1 1 1 z, Rzxy BB21 1 1 1 y, Ryxz BB31 1 1 1 x, Rxyz D3 (32) E 2C33C2 A11 1 1 x2 + y2, z2A21 1 1 z, Rz E 2 1 0 (x, y)(Rx,, Ry) (x2 y2, 2xy) (xz, yz) D4 (422) E 2C4224(CC=)

7 2C2'2C2" A11 1 1 1 1 x2 + y2, z2A21 1 1 1 1 z, Rz BB11 1 1 1 1 x2 y2BB21 1 1 1 1 xy E 2 0 2 0 0 (x, y)(Rx, Ry) (xz, yz) D5E 2C5252C 5C2 A11 1 1 1 x2 + y2, z2A21 1 1 1 z, Rz E12 2 cos 72 2 cos 144 0 (x, y)(Rx, Ry) (xz, yz) E22 2 cos 144 2 cos 72 0 (x2 y2, 2xy) D6 (622) E 2C62C3C223C 23C A11 1 1 1 1 1 x2 + y2, z2A21 1 1 1 1 1 z, Rz BB11 1 1 1 1 1 BB21 1 1 1 1 1 E12 1 1 2 0 0 (x, y)(Rx, Ry) (xz, yz) E22 1 1 2 0 0 (x2 y2, 2xy) Oxford University Press, 2006.

8 All rights reserved. 6 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n 4. The Groups Cn (n = 2, 3, 4, 5, 6) C2 (2mm) E C2 (xz) v (yz) A11 1 1 1 z x2, y2, z2A21 1 1 1 Rzxy BB11 1 1 1 x, Ry xz BB21 1 1 1 y, Rxyz C3 (3m) E 2C33 A11 1 1 z x2 + y2, z2A21 1 1 Rz E 2 1 0 (x, y)(Rx, Ry) (x2 y2, 2xy)(xz, yz) C4 (4mm) E 2C4C22 2 d A11 1 1 1 1 z x2 + y2, z2A21 1 1 1 1 Rz BB11 1 1 1 1 x2 y2BB21 1 1 1 1 xy E 2 0 2 0 0 (x, y)(Rx, Ry) (xz, yz) C5 E 2C5252C 5 A11 1 1 1 z x2 + y2, z2A21 1 1 1 Rz E12 2 cos 72 2 cos 144 0 (x, y)(Rx, Ry) (xz, yz) E22 2 cos 144 2 cos 72 0 (x2 y2, 2xy) C6 (6mm)

9 E 2C62C3C23 3 d A11 1 1 1 1 1 z x2 + y2, z2A21 1 1 1 1 1 Rz BB11 1 1 1 1 1 BB21 1 1 1 1 1 E12 1 1 2 0 0 (x, y)(Rx, Ry) (xz, yz) E22 1 1 2 0 0 (x2 y2, 2xy) Oxford University Press, 2006. All rights reserved. 7 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n 5. The Groups Cnh (n = 2, 3, 4, 5, 6) C2h (2/m) E C2I h Ag1 1 1 1 Rzx2, y2, z2, xy BBg1 1 1 1 Rx, Ryxz, yz Au1 1 1 1 z BBu1 1 1 1 x, y C3h ()6 E C323C hS353S = exp (2 i/3) A' 1 1 1 1 1 1 Rzx2 + y2, z2E' **1111 (x, y) (x2 y2, 2xy) A'' 1 1 1 1 1 1 z E'' **1 11 1 (Rx, Ry) (xz, yz) C4h (4/m)

10 E C4C234C i 34S hS4 Ag1 1 1 1 1 1 1 1 Rzx2 + y2, z2 BBg1 1 1 1 1 1 1 1 (x2 y2, 2xy) Eg1 i 1 i1 i 1 i1i 1 i1i 1 i (Rx, Ry) (xz, yz) Au1 1 1 1 1 1 1 1 z BBu1 1 1 1 1 1 1 1 Eu1i 1i 1i1i1 i 1i 1i1 i (x, y) Oxford University Press, 2006. All rights reserved. 8 Atkins, Child, & Phillips: Tables for Group Theory OXFORD H i g h e r E d u c a t i o n 5. The Groups Cnh (n = 2, 3, 4, 5, 6) ( ) C5hE C525C 35C 45C hS575S 35S 95S = exp(2 i/5) A 1 1 1 1 1 1 1 1 1 1 Rzx2+y2, z21E 2*2*2*2**2 2*2 21111 (x, y) 2E 2**22**2*2*2*2*21111 z (x2 y2, 2xy) A 1 1 1 1 1 1 1 1 1 1 1E 2*2*2 *2 **2 2*221111 (Rx, Ry) (xz, yz) 2E 2**22**2*2*2*2*21111 C6h (6/m) E C6C3C223C 56C i 53S 56S hS6S3 = exp(2 i/6) Ag1 1 1 1 1 1 1 1 1 1 1 1 x2+y2, z2 BBg1 1 1 1 1 1 1 1 1 1 1 1 (Rx, Ry) (xz, yz)